This paper answers to some questions that remained open for some
time in the community of mathematicians working on quasiconformal mapping
theory in subriemannian geometry. The first result presented here is the
characterisation of the rigidity of Carnot groups in the class of C2
contact maps, obtained by extending Tanaka theory from its classical domain of
C∞ contact vector fields to the pseudogroup of local C2
contact mappings.
The second result is a Liouville type theorem proved for all Carnot groups
other than R or R2. The proof rests upon recent results of Capogna
and Cowling and classical results on prolonging the conformal Lie algebra. As
an additional goal, this article aims to provide a partial exposition on Tanaka
theory and to give an elementary proof of a key result due to Guillemin, Quillen
and Sternberg concerning complex characteristics.